$L^2$-Harmonic 1-Forms on Complete Manifolds

Abstract

We study the global behavior of complete minimal $\delta$-stable hypersurfaces in $\mathbf{R}^{n+1}$ by using $L^2$-harmonic 1-forms. We show that a complete minimal $\delta$-stable $\bigg(\delta>\dfrac{(n-1)^2}{n^2}\bigg)$ hypersurface in $\mathbf{R}^{n+1}$ has only one end. We also obtain two vanishing theorems of complete noncompact quaternionic manifolds satisfying the weighted Poincaré inequality. These results are improvements of the first author's theorems on hypersurfaces and quaternionic Kähler manifolds.